| Weakly Lindelof determined Banach spaces not containing $\ell^1(N)$ - Spiros A. Argyros The class of countably intersected families of sets is defined. For any such family we define a Banach space not containing $\ell^{1}(\NN )$. Thus we obtain counterexamples to certain questions related to the heredity problem for W.C.G. Banach spaces. Among them we give a subspace of a W.C.G. Banach space not containing $\ell^{1}(\NN )$ and not being itself a W.C.G. space. Downloads: 11 | |

| A $c_0$ saturated Banach space with tight structure - Spiros A. Argyros It is shown that variants of the HI methods could yield objects closely connected to the classical Banach spaces. Thus we present a new $c_0$ saturated space, denoted as $\mathfrak{X}_0$, with rather tight structure. The space $\mathfrak{X}_0$ is not embedded into a space with an unconditional basis and its complemented subspaces have the following structure. Everyone is either of type I, namely, contains an isomorph of $\mathfrak{X}_0$ itself or else is isomorphic to a subspace of $c_0$ (type I... Downloads: 2 | |

| Banach Spaces Of The Type Of Tsirelson - Spiros A. Argyros To any pair ( M , theta ) where M is a family of finite subsets of N compact in the pointwise topology, and 0{1/n} then T_M^theta is reflexive. Moreover, if the Cantor-Bendixson index of M is greater than omega then T_M^theta does not contain any l^p, while if the Cantor-Bendixson index of M is finite thenT_M^theta contains some l^p or c_o . In particular, if M ={ A subset N : |A| leq n } and {1/n} Downloads: 2 | |

| On spaces admitting no $\ell_p$ or $c_0$ spreading model - Spiros A. Argyros It is shown that for each separable Banach space $X$ not admitting $\ell_1$ as a spreading model there is a space $Y$ having $X$ as a quotient and not admitting any $\ell_p$ for $1 \leq p < \infty$ or $c_0$ as a spreading model. We also include the solution to a question of W.B. Johnson and H.P. Rosenthal on the existence of a separable space not admitting as a quotient any space with separable dual. Downloads: 2 | |

| Examples of k-iterated spreading models - Spiros A. Argyros It is shown that for every $k\in\mathbb{N}$ and every spreading sequence $\{e_n\}_{n\in\mathbb{N}}$ that generates a uniformly convex Banach space $E$, there exists a uniformly convex Banach space $X_{k+1}$ admitting $\{e_n\}_{n\in\mathbb{N}}$ as a $k+1$-iterated spreading model, but not as a $k$-iterated one. Downloads: 1 | |

| Non separable reflexive spaces admitting $\ell_1$ as a unique spreading model - Spiros A. Argyros Examples of non separable reflexive Banach spaces $\mathfrak{X}_{2^{\aleph_0}}$, admitting only $\ell_1$ as a spreading model, are presented. The definition of the spaces is based on $\alpha$-large, $\alpha Downloads: 1 | |

| A reflexive HI space with the hereditary Invariant Subspace Property - Spiros A. Argyros A reflexive hereditarily indecomposable Banach space $\mathfrak{X}_{_{^\text{ISP}}}$ is presented, such that for every $Y$ infinite dimensional closed subspace of $\mathfrak{X}_{_{^\text{ISP}}}$ and every bounded linear operator $T:Y\rightarrow Y$, the operator $T$ admits a non-trivial closed invariant subspace. Downloads: 15 | |

| Interpolating hereditarily indecomposable Banach spaces - Spiros A. Argyros It is shown that every Banach space either contains $\ell ^1$ or it has an infinite dimensional closed subspace which is a quotient of a H.I. Banach space.Further on, $L^p(\lambda )$, $1 Downloads: 8 | |

| Non dentable sets in Banach spaces with separable dual - Spiros A. Argyros A non RNP Banach space E is constructed such that $E^{*}$ is separable and RNP is equivalent to PCP on the subsets of E. Downloads: 4 | |

| The cofinal property of the Reflexive Indecomposable Banach spaces - Spiros A. Argyros It is shown that every separable reflexive Banach space is a quotient of a reflexive Hereditarily Indecomposable space, which yields that every separable reflexive Banach is isomorphic to a subspace of a reflexive Indecomposable space. Furthermore, every separable reflexive Banach space is a quotient of a reflexive complementably $\ell_p$ saturated space with $1 Downloads: 7 | |

| Examples of asymptotically \ell_^1 Banach spaces - Spiros A. Argyros Two examples of asymptotic $\ell_{1}$ Banach spaces are given. The first, $X_{u}$, has an unconditional basis and is arbitrarily distortable. The second, $X$, does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson. We thus answer a question raised by W.T.Gowers. Downloads: 8 | |

| A hereditarily indecomposable L_\infty-space that solves the scalar-plus-compact problem - Spiros A Argyros We construct a hereditarily indecomposable Banach space with dual isomorphic to $\ell_1$. Every bounded linear operator on this space has the form $\lambda I+K$ with $\lambda$ a scalar and $K$ compact. Downloads: 2 | |

| Convex unconditionality and summability of weakly null sequences - Spiros A. Argyros It is proved that every normalized weakly null \sq\ has a sub\sq\ which is convexly unconditional. Further, an Hierarchy of summability methods is introduced and with this we give a complete classification of the complexity of weakly null \sq s. Downloads: 2 | |

| A weak Hilbert space with few symmetries - Spiros A. Argyros We construct a weak Hilbert Banach space such that for every block subspace $Y$ every bounded linear operator on Y is of the form D+S where S is a strictly singular operator and D is a diagonal operator. We show that this yields a weak Hilbert space whose block subspaces are not isomorphic to any of their proper subspaces. Downloads: 1 | |

| A classification of separable Rosenthal compacta and its applications - Spiros A. Argyros The present work consists of three parts. In the first one we determine the prototypes of separable Rosenthal compacta and we provide a classification theorem. The second part concerns an extension of a theorem of S. Todorcevic. The last one is devoted to applications. Downloads: 3 | |

| Hereditarily Indecomposable Banach algebras of diagonal operators - Spiros A. Argyros We provide a characterization of the Banach spaces $X$ with a Schauder basis $(e_n)_{n\in\mathbb{N}}$ which have the property that the dual space $X^*$ is naturally isomorphic to the space $\mathcal{L}_{diag}(X)$ of diagonal operators with respect to $(e_n)_{n\in\mathbb{N}}$ . We also construct a Hereditarily Indecomposable Banach space ${\mathfrak X}_D$ with a Schauder basis $(e_n)_{n\in\mathbb{N}}$ such that ${\mathfrak X}^*_D$ is isometric to $\mathcal{L}_{diag}({\mathfrak X}_D)$ with these B... Downloads: 2 | |

| Modified mixed Tsirelson spaces - Spiros A. Argyros We study the modified and boundedly modified mixed Tsirelson spaces $T_M[({\cal F}_{k_n},\theta_n)_{n=1}^{\infty }]$ and $T_{M(s)}[({\cal F}_{k_n},\theta_n)_{n=1}^{\infty }]$ respectively, defined by a subsequence $({\cal F}_{k_n})$ of the sequence of Schreier families $({\cal F}_n)$. These are reflexive asymptotic $\ell_1$ spaces with an unconditio- nal basis $(e_i)_i$ having the property that every sequence $\{ x_i\}_{i=1}^n$ of normalized disjointly supported vectors contained in $\langle e_i... Downloads: 3 | |

| Saturated extensions, the attractors method and Hereditarily James Tree Space - Spiros A. Argyros In the present work we provide a variety of examples of HI Banach spaces containing no reflexive subspace and we study the structure of their duals as well as the spaces of their linear bounded operators. Our approach is based on saturated extensions of ground sets and the method of attractors. Downloads: 7 | |

| Strictly singular non-compact diagonal operators on HI spaces - Spiros A. Argyros We construct a Hereditarily Indecomposable Banach space $\eqs_d$ with a Schauder basis \seq{e}{n} on which there exist strictly singular non-compact diagonal operators. Moreover, the space $\mc{L}_{\diag}(\eqs_d)$ of diagonal operators with respect to the basis \seq{e}{n} contains an isomorphic copy of $\ell_{\infty}(\N)$. Downloads: 3 | |